Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,33)(29,43)(30,31)(32,38)(34,44)(36,52)(37,51)(39,46)(40,56)(41,50)(42,49)(45,53)(47,55)(48,54);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(27,36)(28,39)(34,43)(38,53)(41,42)(47,48)(51,56);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,38,30,36,44,53)(26,32,34,35,31,45,43,52)(28,40,42,54,39,37,41,55)(33,48,50,56,46,47,49,51);(2,6)(7,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,37)(29,42)(30,39)(31,46)(32,47)(34,50)(35,51)(36,40)(38,55)(41,44)(43,49)(45,48)(52,56)(53,54);(1,4,5,3)(2,8,6,7)(10,16)(11,18)(15,24)(17,19)(25,29,30,44)(26,34,31,43)(27,38,36,53)(28,41,39,42)(32,35,45,52)(33,49,46,50)(37,54,40,55)(47,56,48,51)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=fgi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6956147);